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首页> 外文期刊>Proceedings of the American Mathematical Society >ON THE pTH VARIATION OF A CLASS OF FRACTAL FUNCTIONS
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ON THE pTH VARIATION OF A CLASS OF FRACTAL FUNCTIONS

机译：ON THE pTH VARIATION OF A CLASS OF FRACTAL FUNCTIONS

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摘要

The concept of the pth variation of a continuous function f along a refining sequence of partitions is the key to a pathwise Ito integration theory with integrator f. Here, we analyze the pth variation of a class of fractal functions, containing both the Takagi-van der Waerden and Weierstra beta functions. We use a probabilistic argument to show that these functions have linear pth variation for a parameter p >= 1, which can be interpreted as the reciprocal Hurst parameter of the function. It is shown, moreover, that if functions are constructed from (a skewed version of) the tent map, then the slope of the pth variation can be computed from the pth moment of a (non-symmetric) infinite Bernoulli convolution. Finally, we provide a recursive formula of these moments and use it to discuss the existence and non-existence of a signed version of the pth variation, which occurs in pathwise It (o) over cap calculus when p >= 3 is an odd integer.

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